Problem: Simplify and expand the following expression: $ \dfrac{4y}{5y + 10}+\dfrac{y}{y - 6} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5y + 10)(y - 6)$ Multiply the first term by $\dfrac{y - 6}{y - 6}$ $ \begin{align*} \dfrac{4y}{5y + 10} \times \dfrac{y - 6}{y - 6} & = \dfrac{(4y)(y - 6)}{(5y + 10)(y - 6)} \\ & = \dfrac{4y^2 - 24y}{(5y + 10)(y - 6)}\end{align*} $ Multiply the second term by $\dfrac{5y + 10}{5y + 10}$ $ \begin{align*} \dfrac{y}{y - 6} \times \dfrac{5y + 10}{5y + 10} & = \dfrac{(y)(5y + 10)}{(y - 6)(5y + 10)} \\ & = \dfrac{5y^2 + 10y}{(y - 6)(5y + 10)}\end{align*} $ Now we have: $ = \dfrac{4y^2 - 24y}{(5y + 10)(y - 6)} + \dfrac{5y^2 + 10y}{(y - 6)(5y + 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4y^2 - 24y + 5y^2 + 10y}{(5y + 10)(y - 6)} $ $ = \dfrac{9y^2 - 14y}{(5y + 10)(y - 6)}$ Expand the denominator: $ = \dfrac{9y^2 - 14y}{5y^2 - 20y - 60}$